| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 338.1-a1 |
338.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{2} \) |
$1.87704$ |
$(-a+2), (13)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1.881449080$ |
$15.03051927$ |
2.565537192 |
\( -\frac{182077668135}{104} a - \frac{891994414589}{208} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -8 a - 4\) , \( 7 a + 10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-8a-4\right){x}+7a+10$ |
| 338.1-a2 |
338.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{24} \cdot 13^{6} \) |
$1.87704$ |
$(-a+2), (13)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$0.627149693$ |
$1.670057697$ |
2.565537192 |
\( \frac{5204205755}{4499456} a - \frac{38040813581}{8998912} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 32 a - 99\) , \( 191 a - 426\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(32a-99\right){x}+191a-426$ |
| 338.1-b1 |
338.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{18} \cdot 13^{2} \) |
$1.87704$ |
$(-a+2), (13)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$12.10583107$ |
4.942184841 |
\( -\frac{10730978619193}{6656} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 9191 a - 22515\) , \( -749100 a + 1834913\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(9191a-22515\right){x}-749100a+1834913$ |
| 338.1-b2 |
338.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{6} \cdot 13^{6} \) |
$1.87704$ |
$(-a+2), (13)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$12.10583107$ |
4.942184841 |
\( -\frac{10218313}{17576} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -91 a - 220\) , \( 1444 a + 3538\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-91a-220\right){x}+1444a+3538$ |
| 338.1-b3 |
338.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{2} \) |
$1.87704$ |
$(-a+2), (13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$12.10583107$ |
4.942184841 |
\( \frac{12167}{26} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -9 a + 25\) , \( 40 a - 97\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-9a+25\right){x}+40a-97$ |
| 338.1-c1 |
338.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{24} \cdot 13^{6} \) |
$1.87704$ |
$(-a+2), (13)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$1.962517524$ |
3.204777697 |
\( -\frac{5204205755}{4499456} a - \frac{38040813581}{8998912} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 450 a - 1104\) , \( 12656 a - 31000\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(450a-1104\right){x}+12656a-31000$ |
| 338.1-c2 |
338.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{2} \) |
$1.87704$ |
$(-a+2), (13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$1.962517524$ |
3.204777697 |
\( \frac{182077668135}{104} a - \frac{891994414589}{208} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 291 a + 712\) , \( -290 a - 712\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(291a+712\right){x}-290a-712$ |
| 338.1-d1 |
338.1-d |
$2$ |
$7$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{14} \) |
$1.87704$ |
$(-a+2), (13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$0.385597965$ |
1.101937971 |
\( -\frac{1064019559329}{125497034} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -213\) , \( -1257\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-213{x}-1257$ |
| 338.1-d2 |
338.1-d |
$2$ |
$7$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{14} \cdot 13^{2} \) |
$1.87704$ |
$(-a+2), (13)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$18.89430030$ |
1.101937971 |
\( -\frac{2146689}{1664} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -3\) , \( 3\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-3{x}+3$ |
| 338.1-e1 |
338.1-e |
$2$ |
$7$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{14} \) |
$1.87704$ |
$(-a+2), (13)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.6.3 |
$1$ |
\( 2 \cdot 7 \) |
$0.089317794$ |
$3.254622356$ |
1.661464272 |
\( -\frac{1064019559329}{125497034} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 4254 a - 10420\) , \( -263508 a + 645460\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(4254a-10420\right){x}-263508a+645460$ |
| 338.1-e2 |
338.1-e |
$2$ |
$7$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{14} \cdot 13^{2} \) |
$1.87704$ |
$(-a+2), (13)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.6.1 |
$1$ |
\( 2 \) |
$0.625224564$ |
$3.254622356$ |
1.661464272 |
\( -\frac{2146689}{1664} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -53 a - 127\) , \( -592 a - 1451\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-53a-127\right){x}-592a-1451$ |
| 338.1-f1 |
338.1-f |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{24} \cdot 13^{6} \) |
$1.87704$ |
$(-a+2), (13)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$0.627149693$ |
$1.670057697$ |
2.565537192 |
\( -\frac{5204205755}{4499456} a - \frac{38040813581}{8998912} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -32 a - 99\) , \( -191 a - 426\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-32a-99\right){x}-191a-426$ |
| 338.1-f2 |
338.1-f |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{2} \) |
$1.87704$ |
$(-a+2), (13)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1.881449080$ |
$15.03051927$ |
2.565537192 |
\( \frac{182077668135}{104} a - \frac{891994414589}{208} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 8 a - 4\) , \( -7 a + 10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(8a-4\right){x}-7a+10$ |
| 338.1-g1 |
338.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{18} \cdot 13^{2} \) |
$1.87704$ |
$(-a+2), (13)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$7.652505194$ |
$0.265819283$ |
1.660903829 |
\( -\frac{10730978619193}{6656} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -460\) , \( -3830\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-460{x}-3830$ |
| 338.1-g2 |
338.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{6} \cdot 13^{6} \) |
$1.87704$ |
$(-a+2), (13)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$2.550835064$ |
$2.392373550$ |
1.660903829 |
\( -\frac{10218313}{17576} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -5\) , \( -8\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-5{x}-8$ |
| 338.1-g3 |
338.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{2} \) |
$1.87704$ |
$(-a+2), (13)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$0.850278354$ |
$21.53136195$ |
1.660903829 |
\( \frac{12167}{26} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}$ |
| 338.1-h1 |
338.1-h |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{2} \) |
$1.87704$ |
$(-a+2), (13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$1.962517524$ |
3.204777697 |
\( -\frac{182077668135}{104} a - \frac{891994414589}{208} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -288 a + 715\) , \( 1002 a - 2449\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-288a+715\right){x}+1002a-2449$ |
| 338.1-h2 |
338.1-h |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{24} \cdot 13^{6} \) |
$1.87704$ |
$(-a+2), (13)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$1.962517524$ |
3.204777697 |
\( \frac{5204205755}{4499456} a - \frac{38040813581}{8998912} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -450 a - 1104\) , \( -12656 a - 31000\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-450a-1104\right){x}-12656a-31000$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
